Counterexamples to Grötzsch–Sachs–Koester's conjecture

Abstract

Let G be a 4-regular planar graph and suppose that G has a cycle decomposition S (i.e., each edge of G is in exactly on cycle of the decomposition) with every pair of adjacent edges on a face always in different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Grötzsch–Sachs–Koester's conjecture states that if the cycles of G can be partitioned into four classes, such that two cycles in the same classes are disjoint, G is vertex 3-colorable. In this note, the conjecture is disproved.